<P> There is a growing interest in magnesium and its alloys due to their high strength to weight ratio. Magnesium is of particular interest to the automotive industry as a consequence of the current pressure to reduce green house gas emissions from the transportation sector through vehicle weight reduction. However, there is a lack of knowledge concerning the formability of magnesium. As a result, the application of magnesium as a commercial material has not been fully exploited. Much has been learned from the constitutive modeling of materials such as aluminum and steel. Therefore, this thesis considers the constitutive modeling of magnesium and its alloys. </p> <p> Based on this motivation, polycrystal plasticity theories that have been established and used to characterize aluminum and steel are studied. The validity of these theories is examined with respect to magnesium and its alloys. The magnesium system is composed of the hexagonal closed-packed (HCP) crystal structure. Therefore, a strong plastic anisotropy is induced in magnesium crystals due to the limited number of slip systems that may be activated with ease. The models proposed by Taylor and Sachs neglect strain and stress heterogeneities respectively. As a result, the models are either too stiff or too soft to study magnesium due to the anisotropic nature of the crystal structure. The intermediate models; self-consistent models, which are able to consider the heterogeneities among the grains in polycrystals, are believed to be more suitable to study magnesium and its alloys. Therefore, a large strain elastic-viscoplastic self-consistent (EVPSC) model is developed for polycrystalline materials. Both rate sensitive slip and twinning are included as mechanisms of plastic deformation, while elastic anisotropy is accounted for in the elastic modulus. The transition from single crystal plasticity to polycrystal plasticity is based on a completely self-consistent approach. It is shown that the differences in the predicted stress-strain curves and texture evolutions based on the EVPSC and the viscoplastic self-consistent (VPSC) model proposed by Lebensohn and Tome (1993) are negligible at large strains for monotonic loadings. For the deformations involving unloading and strain path changes, the EVPSC predicts a smooth elasto-plastic transition, while the VPSC model gives a discontinuous response because the model is incapable of modeling elastic deformation. In addition, it is demonstrated that the EVPSC model can capture some important experimental features which cannot be simulated by using the VPSC model. </p> <p> Various self-consistent schemes exist for EVPSC and VPSC models. However, the evaluations of these models are not complete. Therefore, an examination of various polycrystal plasticity models is made, based on comparisons of the predicted and experimental stress responses as well as the R values, to assess their validity. It is established that, among the models examined, the self-consistent models with grain interaction stiffuess values halfway between those of the limiting Secant (stiff) and Tangent (compliant) approximations give the best results. Among the available options, the Affine self-consistent scheme results in the best overall performance. Furthermore, it is demonstrated that the R values under uniaxial tension and compression within the sheet plane show a strong dependence on the imposed strain. This suggests that the development of anisotropic yield functions using measured R values, must account for the strain. dependence. </p> <p> The recently developed large strain elastic visco-plastic self-consistent (EVPSC) model, which incorporates both slip and twinning deformation mechanisms, is used to study .the lattice strain evolution in extruded magnesium alloy AZ31 under uniaxial tension and compression. The results are compared against in-situ neutron diffraction measurements done on the same alloy. For the first time, the effects of stress relaxation and strain creep on lattice strain measurements in respectively displacement controlled and load controlled in-situ tests are numerically assessed. It is found that the stress relaxation, has a significant effect on the lattice strain measurements. It is also observed that although the creep does not significantly affect the trend of the lattice strain evolution, a better agreement with the experiments is found if creep is included in the simulations. </p> <p> In conjunction with the M-K approach developed by Marciniak and Kuczynski (1967), the EVPSC model is applied to study the sheet metal formability of magnesium alloys in terms of the forming limit diagram (FLO). The role of crystal plasticity models and the effects of basal texture on formability of magnesium alloy AZ31 B sheet are studied numerically. It is observed that formability in HCP polycrystalline materials is very sensitive to the intensity of the basal texture. The path-dependency of formability is examined based on different non-proportional loading histories, which are combinations of two linear strain paths. It is found that while the FLO in strain space is very sensitive to strain path changes, the forming limit stress diagram (FLSO) in stress space is much less path-dependent. It is suggested that the FLSO is much more favourable than the FLO in representing forming limits in the numerical simulation of sheet metal forming processes. The numerical results are found to be in good qualitative agreement with experimental observations. </p> / Thesis / Doctor of Philosophy (PhD)

This DPhil project is set in the context of single crystal elasticity-plasticity finite element modelling. Its core objective was to develop and implement a methodology for predicting the evolution of texture in single and dual-phase material systems. This core objective has been successfully achieved. Modelling texture evolution entails essentially modelling large deformations (as accurately as possible) and taking account of the deformation mechanisms that cause texture to change. The most important deformation mechanisms are slip and twinning. Slip has been modelled in this project and care has been taken to explore conditions where it is the dominant deformation mechanism for the materials studied. Modelling slip demands that one also models dislocations since slip is assumed to occur by the movement of dislocations. In this project a model for geometrically necessary dislocations has been developed and validated against experimental measurements. A texture homogenisation technique which relies on interpretation of EBSD data in order to allocate orientation frequencies based on representative area fractions has been developed. This has been coupled with a polycrystal plasticity RVE framework allowing for arbitrarily sized RVEs and corresponding allocation of crystallographic orientation. This has enabled input of experimentally measured initial textures into the CPFE model allowing for comparison of predictions against measured post-deformation textures, with good agreement obtained. The effect of texture on polycrystal physical properties has also been studied. It has been confirmed that texture indeed has a significant role in determining the average physical properties of a polycrystal. The thesis contributes to the following areas of micro-mechanics materials research: (i) 3D small deformation crystal plasticity finite element (CPFE) modelling, (ii) geometrically necessary dislocation modelling, (iii) 3D large deformation CPFE modelling, (iv) texture homogenisation methods, (v) single and dual phase texture evolution modelling, (vi) prediction of polycrystal physical properties, (vii) systematic calibration of the power law for slip based on experimental data, and (viii) texture analysis software development (pole figures and Kearns factors).